The Bible speaks of "false brethren" in 2 Cor 11:26 and Gal 2:4 (NASB). They are not saved, but have put on a false front. That is different than those who think they are saved because of what they do, such as as the crowd who appealed to Jesus on the basis of their works.
The OP properly left out the key regarding whether one is a true convert or a false convert. Faith. The OP did not mention what they believed, but only what they said and did.
To say "I'm saved/a Christian" is not the same thing as being one.
So, a "false convert" is one who thinks they are a Christian, but have never placed their trust and faith in Jesus Christ for eternal life.
Regarding the "inward" stuff, no human can see the new birth. We can only see deeds, which may or may not validate the new birth. There are many Mormons whose lives are better than a lot of evangelicals, as an example.
The ONLY people are are truly saved are those who have placed their faith and trust solely in Jesus Christ for eternal life. No one else. Regardless of what they say or do.